Spontaneous Gravity Wave Radiation in a Shallow Water System on a Rotating Sphere

Abstract

The effect of the earth's rotation on spontaneous inertial gravity wave radiation from an unsteady rotational flow is investigated numerically using a shallow water system on a rotating sphere, changing three parameters, the Rossby number (1 ≤ Ro ≤ 30), the Froude number (0.1 ≤ Fr ≤ 0.7), and the latitudinal positions of the jet (11.25 • N ≤ θ 0 ≤ 78.75 • N). The spectral-like three-point combined compact difference (sp-CCD) scheme is used for numerical calculation to estimate the amplitude of gravity waves with high accuracy and resolution similar to the spherical harmonics model. The jet, which is initially balanced but evolves with time because of barotropic instability, is maintained by relaxation forcing. The time variations of nearly balanced rotational flows continuously radiate gravity waves. While the amount of gravity wave flux for large Ro (≥ 10) in all positions of the jet is almost constant, that for relatively small Ro (< 10) depends considerably on the positions of the jet and the observational latitude. First, there is almost no gravity wave radiation from the jet positioned at a high latitude. Second, even when gravity waves are radiated from the jet positioned at low latitude, they rarely propagate toward a higher latitude. To discuss the results, the source of gravity waves is introduced as an analogy to the aeroacoustic sound wave radiation theory (Lighthill theory) with an f -plane approximation at the position of the jet. Spectral analysis of the sources reasonably explains the waves. Because the effect of the earth's rotation varies with the latitude, gravity waves are radiated only from an unsteady source with higher frequency than the inertial cut-off frequency at the position of the jet. In a similar way, gravity waves that propagate toward high latitudes should have a higher frequency than the inertial cut-off frequency at that latitude.

Inertia-gravity wave radiation from the merging of two co-rotating vortices in the f-plane shallow water system

“…5 While gravity wave radiation in a shallow water system is different from that in a continuously stratified one, 6 their fundamental study in geophysical fluid dynamics is nevertheless important. 3 Spontaneous gravity wave radiation from unsteady vortical flows in rotating shallow water has been investigated 5,[7][8][9] and cyclone-anticyclone asymmetry has recently been found. 10 However, it is not evident whether this asymmetry appears in complicated vortical flows, since analysis and numerical verification were performed only for an idealized case of a co-rotating point vortex pair.…”

“…5,9 This quantity is conserved when gravity waves propagate into the far field. The maximum value of the gravity wave flux averaged in the θ direction at r = 45 is shown for several Ro values (0.5 ≤ Ro ≤ ∞) in Fig.…”

“…10. Spontaneous gravity wave radiation in rotating shallow water is analogous to sound emission from a vortex (Lighthill-Ford theory 6,18 ); the quadrupole source of waves is approximately expressed using the unsteady motions of vortical flows, In addition, the source originating in Coriolis acceleration is important at smaller Ro, [8][9][10] which is expressed as…”

Nonlinear Interaction Between Vortex and Wave in Rotating Shallow Water

“…For the rotating case, on the other hand, the second term on the right-hand side of Eq. (5) becomes also important for relatively larger f [20,21]. Then, for the relevant term on the righthand side of Eq.…”

“…As an extension of Ford's works [12,13], several numerical works are performed in shallow water system on an f-plane [19,20] and a sphere [21]. While it is widely known that the effect of the Earth's rotation suppresses inertia-gravity wave radiation, previous studies [20,21] have reported that the effect of the Earth's rotation intensify gravity wave radiation in some parameter space. In this chapter, recent results of the inertia-gravity wave radiation from nearly balanced vortical flows as an extension of sound wave generation from vortical flows are reviewed.…”